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Identifier

St001566: Permutations ⟶ ℤ

Values

[1] => 1

[1,2] => 2

[2,1] => 2

[1,2,3] => 3

[1,3,2] => 2

[2,1,3] => 2

[2,3,1] => 2

[3,1,2] => 2

[3,2,1] => 3

[1,2,3,4] => 4

[1,2,4,3] => 3

[1,3,2,4] => 2

[1,3,4,2] => 2

[1,4,2,3] => 3

[1,4,3,2] => 3

[2,1,3,4] => 3

[2,1,4,3] => 2

[2,3,1,4] => 3

[2,3,4,1] => 3

[2,4,1,3] => 2

[2,4,3,1] => 2

[3,1,2,4] => 2

[3,1,4,2] => 2

[3,2,1,4] => 3

[3,2,4,1] => 3

[3,4,1,2] => 2

[3,4,2,1] => 3

[4,1,2,3] => 3

[4,1,3,2] => 3

[4,2,1,3] => 2

[4,2,3,1] => 2

[4,3,1,2] => 3

[4,3,2,1] => 4

[1,2,3,4,5] => 5

[1,2,3,5,4] => 4

[1,2,4,3,5] => 3

[1,2,4,5,3] => 3

[1,2,5,3,4] => 4

[1,2,5,4,3] => 3

[1,3,2,4,5] => 3

[1,3,2,5,4] => 3

[1,3,4,2,5] => 3

[1,3,4,5,2] => 3

[1,3,5,2,4] => 3

[1,3,5,4,2] => 3

[1,4,2,3,5] => 3

[1,4,2,5,3] => 3

[1,4,3,2,5] => 3

[1,4,3,5,2] => 3

[1,4,5,2,3] => 3

[1,4,5,3,2] => 3

[1,5,2,3,4] => 4

[1,5,2,4,3] => 3

[1,5,3,2,4] => 2

[1,5,3,4,2] => 2

[1,5,4,2,3] => 3

[1,5,4,3,2] => 4

[2,1,3,4,5] => 4

[2,1,3,5,4] => 3

[2,1,4,3,5] => 3

[2,1,4,5,3] => 2

[2,1,5,3,4] => 3

[2,1,5,4,3] => 3

[2,3,1,4,5] => 4

[2,3,1,5,4] => 3

[2,3,4,1,5] => 4

[2,3,4,5,1] => 4

[2,3,5,1,4] => 3

[2,3,5,4,1] => 3

[2,4,1,3,5] => 3

[2,4,1,5,3] => 2

[2,4,3,1,5] => 2

[2,4,3,5,1] => 2

[2,4,5,1,3] => 2

[2,4,5,3,1] => 3

[2,5,1,3,4] => 3

[2,5,1,4,3] => 3

[2,5,3,1,4] => 3

[2,5,3,4,1] => 3

[2,5,4,1,3] => 3

[2,5,4,3,1] => 3

[3,1,2,4,5] => 3

[3,1,2,5,4] => 2

[3,1,4,2,5] => 3

[3,1,4,5,2] => 3

[3,1,5,2,4] => 2

[3,1,5,4,2] => 2

[3,2,1,4,5] => 3

[3,2,1,5,4] => 3

[3,2,4,1,5] => 3

[3,2,4,5,1] => 3

[3,2,5,1,4] => 3

[3,2,5,4,1] => 3

[3,4,1,2,5] => 3

[3,4,1,5,2] => 3

[3,4,2,1,5] => 3

[3,4,2,5,1] => 3

[3,4,5,1,2] => 3

[3,4,5,2,1] => 3

[3,5,1,2,4] => 2

[3,5,1,4,2] => 2

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Description

The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$

Code

def arith_prog_next(pi,lst):
    new_lst = []
    for seq in lst:
        for i in range(seq[-1]+1,len(pi)):
            if pi[i] - pi[seq[-1]] == pi[seq[-1]] - pi[seq[-2]]:
                new_lst.append(seq+[i])
    return new_lst

def statistic(pi):
    n = len(pi)
    if n <= 2:
        return n
    stat = 1
    lst = [[i,j] for j in range(n) for i in range(j)]
    while lst:
        lst = arith_prog_next(pi,lst)
        stat += 1
    return stat

Created

Jul 15, 2020 at 08:37 by Christian Stump

Updated

Jul 15, 2020 at 08:37 by Christian Stump